The revolutionary tendencies could have started earlier.
And, where can we find a map of the distribution of European populations in the New World between 1500 and 1800? Wouldn't there be a lot of unoccupied territory?
They might go inland. When Greeley said, "Go West, young man", he wasn't referring to California, he was referring to Illinois! (Although Wikipedia has an article on the phrase of course, and it's complicated...)
There might be a mix of republics and monarchies.
When the king of Portugal reaches Brazil, for example, he could just say, the new capital of my kingdom is now in Brazil, and his subjects are the population of Brazil. The governor of Brazil, presumably appointed by the king, would probably go along with it.
Meanwhile, the revolutionaries in Portugal might not feel like sending a navy to recapture Brazil, they'd rather consolidate their power at home.
I suppose it's stretching credibility for the king of Poland to settle in Santiago or wherever. But the nobility can be awfully stubborn, and maybe some sycophant claims to have a map to El Dorado so the sycophant can keep his position and get a free ride to the New World.
Another possibility is much of the aristocracy going over there but not all, and not all the kings.
Does that help any?
Thanks.